The classes that I was scheduled to teach in the Fall were Math 406 (Introduction to Number Theory) and Math 274 (History of Mathematics). […]

I have taught Math 274 quite a few times and I have become convinced that the course is misclassified; if you want it to be a 200 series course then it should be changed to “A History of Mathematicians”, while the “History of Mathematics” should be a 400 level course and reserved for students who have had a fair amount of what might be called “Pure Mathematics”; Math 406 would be an ideal prerequisite.

Let me explain my reasoning. By having it as a 200 level course and with “History” as part of the title, many students, specially those whose arithmetical skills are well below that of the ancient egyptian scribes, see this as a course as a simple course in which they can easily get an “A” or “B” by simply memorizing names, dates and events. Now although knowing that Girolamo Cardamo (Cardan) was found guilty of heresy and was going to be burnt at the stake (he had cast a horoscope of Jesus Christ—the heresy being that it implied that God was controlled by the Stars and thus God was not omnipotent—) is in itself an interesting event in the life of Cardamo and also gives us an insight into the social mores of his times; it gives us very little information about the development of Mathematics. In other words it belongs much more to a “History of Mathematicians” than to a “History of Mathematics”! What would pertain to the “History of Mathematics” would be an analysis of the method by which Cardamo solved the cubic equation and that completely overwhelms the marginal student.

But the problem is deeper than simply of not understanding the method. To the student who has never tried to “manually” (i.e. without using a graphing calculator or computer) solve a cubic equation the whole operation appears to be a waste of time and the student then complains that the course is not a “History of Mathematics” but rather an advanced course on “Number Theory”.

So what should be the principal content of a “History of Mathematics” course? A clue can be found in alternate names that have been used for Mathematics, namely “Science of Number” and “Science of Measurement”. I doubt that anyone would disagree that the beginning of Mathematics is intimately connected with the beginning of Number and thus that an “(Early) History of Mathematics” is in fact a “History of Number”.

In fact I venture to suggest that Mathematics begins when Homo Sapiens replaces “number of …..” by simply “number”, even though at the time there was not a clearly defined concept of “number” and there was a great deal of confusion between “number”, “numeral” and “number word” (it was historically much later that it was realized (postulated) that the number two was what was common to (i) a pair of doves, (ii) a brace of hounds, (iii) a yoke of oxen …) . In any case, starting from the concept of “number of …..”, the Greeks were able to justify the Rational Numbers. But what are the grounds for calling the (positive) square root of two a “number”? Or for that matter, when can a mathematical object be called a “number”? In my view the best answer to “What is a number” was given by R. Dedekind [1888]:

” The numbers are free creations of man’s mind, they serve as a means of apprehending the difference of things more easily and more sharply. ”

(I would go as far as to replace “number” by “mathematics”.)

It is clear that a student who had not experienced, or at the very least is aware of, some of the wonderful achievements of Mathematics, would be completely bored in such a course.

Thus my recommendations for Math 274 (History of Mathematics) is that it be returned to the History Department–where it would become a “(Social) History of Mathematics” and that the Mathematical “History of Mathematics” become either a 400 level course, with say Math 406 as a prerequisite, or else an undergraduate seminar–in which the students get appropriate credits towards graduation–.